England looked like drawing level with the defending champions for the second time when Mason Mount was bundled over by Theo Hernandez in the box in the 81st minute.
But Kane — who had scored from the spot earlier in the second half to make it 1-1 before a Olivier Giroud header put France back in front — sprayed his shot high.
The French striker had just been denied by a desperate Jordan Pickford save when a second opportunity presented itself and Giroud made no mistake.
England had fought hard to get back on level terms after a rough first half. Kane converted a 54th minute penalty to make it 1-1 after Bukayo Saka was clipped by Aurelien Tchouameni.
Kane smashed his spot kick into the top left corner of the net to claim justice after both he and Saka were denied by the referees in a controversial first half.
England trailed 1-0 at halftime after a stunning strike by Tchouameni.
France knocked the ball around in the England half before teeing up Tchouameni in the 17th minute, who unleashed a cannon from long-range.
It was the first goal England has conceded from outside of the box since the 2014 World Cup and was France’s first from that distance in this tournament.
But England players were up in arms after believing Saka was fouled in the lead-up to the goal.
The complaints kept coming when Harry Kane went down in the box about 10 minutes later.
It appeared to be a clumsy challenge by Dayot Upamencano, who tugged Kane back before their legs tangled.
But after a long look it was ruled no penalty and England remained frustrated as the first period came to an end.
Heartbreaking once again, but no shame in losing to an excellent French team and congratulations to them. There was so little to choose between the two sides. This fine young @England team gave their all and will only get better. Their time will come.
Absolutely gutted for the lads but proud of the way they’ve played this tournament and they should be proud. Congratulations to @HKane on equalling the record, he’ll be on his own soon and couldn’t think of anyone better to take over. Keep your head up Harry 💪🏼🏴 @England